Wreath Lie Algebras Cristina Di Pietro Cristina Di Pietro 1 Lie - - PowerPoint PPT Presentation

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Lie Algebras, their Classification and Applications University of Trento 25-27 July 2005

Wreath Lie Algebras

Cristina Di Pietro

Cristina Di Pietro –1–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Lie algebras associated with a pro-p-group It is well-known that it is possible to attach a Lie ring L(G) to any pro-p-group G, defining a suitable Lie bracket on the sum of the factors of a strongly central series of G. The properties of the Lie ring depend on the choice of a central series used in this construc- tion.

Cristina Di Pietro –2–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Lie algebras associated with a pro-p-group It is well-known that it is possible to attach a Lie ring L(G) to any pro-p-group G, defining a suitable Lie bracket on the sum of the factors of a strongly central series of G. The properties of the Lie ring depend on the choice of a central series used in this construc- tion. If such factors have exponent p, then the Lie ring turns out to be a Lie algebra over the prime field Fp.

Cristina Di Pietro –2-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Structure of L(G ≀ Cp) Let 2 = p ∈ P; let Cp be the cyclic group of order p, F = Fp , F(ǫ) the divided power algebra and δ its canonical derivation. Let G be a p-group. Then L(G ≀ Cp) depends on L(G). Theorem 1 Let G be a finitely generated (pro-)p-group with γi(G)p ⊆ γi+1(G) for each

i ≥ 1. Then L(G ≀ Cp) ∼ = (L(G) ⊗ F(ǫ)) ⋊ < d >,

where d = idL(G) ⊗ δ is a derivation of order p.

Cristina Di Pietro –3–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Iterated wreath algebras Define the wreath operator w that associates to any Lie algebra L the wreath algebra of L:

w(L) := (L ⊗ F(ǫ)) ⋊ < d >,

where, as above, d = idL ⊗ δ.

Cristina Di Pietro –4–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Iterated wreath algebras Define the wreath operator w that associates to any Lie algebra L the wreath algebra of L:

w(L) := (L ⊗ F(ǫ)) ⋊ < d >,

where, as above, d = idL ⊗ δ. Let W(n) = Cp ≀ · · · ≀ Cp

• n

; by Theorem 1,

L (W(n)) = wn−1(F) := ωn(F) ↑ n-steps wreath algebra

Cristina Di Pietro –4-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

The rˆ

• le of W(n)

W(n) is the Sylow p-subgroup both of Sym(pn) and SL(pn−1(p − 1), Z); by Cayley’s

Theorem, if G is a group of order pn, then

G ֒ → W(n).

Moreover, by Vol’vaˇ cev’s Theorem (1963), if P is a finite p-group with an irreducible representation on the Q-module V , then

P ֒ → W(n)

and V = M|P , where M is the canonical module of W(n).

Cristina Di Pietro –5–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

The rˆ

• le of ωn(F)

Is there an analogous result about modular Lie algebras and ωn(F) ? Yes, there is; the

{ωn(F)}n are the “containers” of the finite-dimensional nilpotent “absolutely irreducible”

linear Lie algebras.

Cristina Di Pietro –6–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

The rˆ

• le of ωn(F)

Is there an analogous result about modular Lie algebras and ωn(F) ? Yes, there is; the

{ωn(F)}n are the “containers” of the finite-dimensional nilpotent “absolutely irreducible”

linear Lie algebras. Definition ρ : L → gl(V ) is an absolutely irreducible representation if it is irreducible over any extension of the base field K, or, equivalently, if it is irreducible and

Cgl(V ) (ρ(L)) = KIdV .

Cristina Di Pietro –6-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

The rˆ

• le of ωn(F)

Is there an analogous result about modular Lie algebras and ωn(F) ? Yes, there is; the

{ωn(F)}n are the “containers” of the finite-dimensional nilpotent “absolutely irreducible”

linear Lie algebras. Definition ρ : L → gl(V ) is an absolutely irreducible representation if it is irreducible over any extension of the base field K, or, equivalently, if it is irreducible and

Cgl(V ) (ρ(L)) = KIdV .

Example ωn(F) has a faithful absolutely irreducible representation on its canonical module.

Cristina Di Pietro –6-b–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Absolutely irreducible case over perfect field Several results about irreducible representations over algebrically closed fields ( [“Modular Lie algebras and their representations”, H.Strade-R.Farnsteiner] ) can be extended to the absolu- tely irreducible case over perfect fields.

Cristina Di Pietro –7–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Absolutely irreducible case over perfect field Several results about irreducible representations over algebrically closed fields ( [“Modular Lie algebras and their representations”, H.Strade-R.Farnsteiner] ) can be extended to the absolu- tely irreducible case over perfect fields. Such results hold for restricted Lie algebras, i.e. for Lie algebras with a p-mapping [p], that is a function satisfying similar properties to z → zp in associative modular algebras.

Cristina Di Pietro –7-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Absolutely irreducible case over perfect field Several results about irreducible representations over algebrically closed fields ( [“Modular Lie algebras and their representations”, H.Strade-R.Farnsteiner] ) can be extended to the absolu- tely irreducible case over perfect fields. Such results hold for restricted Lie algebras, i.e. for Lie algebras with a p-mapping [p], that is a function satisfying similar properties to z → zp in associative modular algebras. However, these results can be extended to non-restricted algebras, because each algebra is embedded in a restricted one, preserving finite-dimensionality, nilpotency and the properties

• f the associated representations.

Cristina Di Pietro –7-b–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Embedding for nilpotent algebras As a consequence, a nilpotent algebra L = I ⊕ Fx, with a maximal p-ideal I having a faithful absolutely irreducible representation (with character S) on W and such that x[p] ∈ I, can be embedded in w(I):

Cristina Di Pietro –8–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Embedding for nilpotent algebras As a consequence, a nilpotent algebra L = I ⊕ Fx, with a maximal p-ideal I having a faithful absolutely irreducible representation (with character S) on W and such that x[p] ∈ I, can be embedded in w(I):

L ֒ → w(I) = (I ⊗ F(ǫ))⋊ < d > x → d + (x[p] + S(x)pIdW ) ⊗ ǫ(p−1) I ∋ y → p−1

i=0 [y,i x] ⊗ ǫ(i)

Cristina Di Pietro –8-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Embedding for nilpotent algebras As a consequence, a nilpotent algebra L = I ⊕ Fx, with a maximal p-ideal I having a faithful absolutely irreducible representation (with character S) on W and such that x[p] ∈ I, can be embedded in w(I):

L ֒ → w(I) = (I ⊗ F(ǫ))⋊ < d > x → d + (x[p] + S(x)pIdW ) ⊗ ǫ(p−1) I ∋ y → p−1

i=0 [y,i x] ⊗ ǫ(i)

W ⊗ F(ǫ) ∼ = IndL

I (W, S) = p−1 i=0 W ⊗ xi

w ⊗ ǫ(i) → w ⊗ xp−1−i

Cristina Di Pietro –8-b–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Main result Let L be a Lie algebra of finite-dimension over a perfect field. If L is a (restricted) nilpotent absolutely irreducible linear algebra on V ,

Cristina Di Pietro –9–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Main result Let L be a Lie algebra of finite-dimension over a perfect field. If L is a (restricted) nilpotent absolutely irreducible linear algebra on V , then there exist k (≤ dim L/C(L)), a

p-subalgebra Q ⊇ C(L) of codimension k and a 1-dimensional Q-submodule M of V

such that

L ֒ → ωk+1(F),

Cristina Di Pietro –9-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

Main result Let L be a Lie algebra of finite-dimension over a perfect field. If L is a (restricted) nilpotent absolutely irreducible linear algebra on V , then there exist k (≤ dim L/C(L)), a

p-subalgebra Q ⊇ C(L) of codimension k and a 1-dimensional Q-submodule M of V

such that

L ֒ → ωk+1(F), V = M ⊗ F(ǫ1) ⊗ · · · ⊗ F(ǫk),

and

dim V = pk.

Cristina Di Pietro –9-c–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

A possible application Let L be a just infinite solvable Lie algebra.

Cristina Di Pietro –10–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

A possible application Let L be a just infinite solvable Lie algebra. By some results of [“Just infinite periodic Lie algebras” Proc. of the Gainesville conference in honour of Thompson - N. Gavioli, V. Monti, C.M. Scoppola] and [D.Young, unpublished note],

• ∃ ! finite-dimensional Lie algebra L such that L ֒

→ Loop(L) = L ⊗ tF[t], with L

minimal with respect to this embedding (the MNUA);

• ∃ ! maximal abelian ideal A (the nilradical) of L such that

L/A ⋉ A → L/I ⋉ I,

where I is the unique minimal ideal of the MNUA, L/I is nilpotent and I is an irreducible L/I-module.

Cristina Di Pietro –10-a–

Lie Algebras, their Classification and Applications, University of Trento- July 2005

A possible application Let L be a just infinite solvable Lie algebra. By some results of [“Just infinite periodic Lie algebras” Proc. of the Gainesville conference in honour of Thompson - N. Gavioli, V. Monti, C.M. Scoppola] and [D.Young, unpublished note],

• ∃ ! finite-dimensional Lie algebra L such that L ֒

→ Loop(L) = L ⊗ tF[t], with L

minimal with respect to this embedding (the MNUA);

• ∃ ! maximal abelian ideal A (the nilradical) of L such that

L/A ⋉ A → L/I ⋉ I,

where I is the unique minimal ideal of the MNUA, L/I is nilpotent and I is an irreducible L/I-module. If I is an absolutely irreducible module, then

L/I ֒ → ωn(F).

A study of the absolutely irreducible subalgebras of ωn(F) gives information on the structure

• f L/A.

Cristina Di Pietro –10-d–